Problem:
Solution:
Part (a) is really just another way of stating the fundamental theorem of algebra. Note that the set $ V(f) $ is simply the set of root of $ f $.
Part (b), GCD is 1 $ \Leftrightarrow $ these polynomial do not have a common monic factor $ \Leftrightarrow $ these polynomial does not have a common root $ \Leftrightarrow $ the variety is the empty set.
Part (c), Compute the GCD, and return the variety is empty if the GCD is 1.
Solution:
Part (a) is really just another way of stating the fundamental theorem of algebra. Note that the set $ V(f) $ is simply the set of root of $ f $.
Part (b), GCD is 1 $ \Leftrightarrow $ these polynomial do not have a common monic factor $ \Leftrightarrow $ these polynomial does not have a common root $ \Leftrightarrow $ the variety is the empty set.
Part (c), Compute the GCD, and return the variety is empty if the GCD is 1.
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